First-Principles calculations of the structural and electronic properties of the high-K dielectric HfO2

Kazuhito Nishitani 1,2, Patrick Rinke 2, Abdallah Qteish 3,Philipp Eggert 2, Javad Hashemifar 2,Peter Kratzer 2, and Matthias Scheffler 2

1 Corporate Manufacturing Engineering Center, Toshiba Corporation2 Fritz-Haber-Institut der Max-Planck-Gesellschaft3 Physics Department, Yarmouk University

Introduction

HfO2

Direct tunneling current Ig = I0 exp (- t )

Fundamental properties about HfO2 by first-principles calculations( structural and electronic properties)

IDSAT * Cox * K / t

IgGate electrode

VgBand offset

ESiIg

t

e-

high-K material with large physical thickness

IDSAT

(1) high dielectric constant ( HfO2 ~25 , SiO2 ~ 4 )(2) wide bandgap ( HfO2 ~6eV, SiO2 ~ 9eV)(3) good thermal stability (amorphous phase)

Scaling of MOS-FET

Transistor SpeedLow power consumptionManufacturing costs

Demand

HfO2 crystallized structure

cubic phase tetragonal phase monoclinic phase

a aa

c

ab

c

a1= (0, a/2, a/2)a2= (a/2, 0, a/2)a3= (a/2, a/2, 0)

a1= (-a/2, a/2, 0)a2= (a/2, a/2, 0)a3= (0, 0, c)

<Unit cell> <Unit cell>

Oxygen

Hafnium

Hafnium : seven-fold coordinated

Oxygen : three-fold coordinated four-fold coordinated

a

a

Outline

1. Pseudo potential and calculation method

2. Structural property (cubic-HfO2, tetragonal-HfO2)

3. Electronic property (cubic-HfO2, tetragonal-HfO2)

4. Comparison between cubic and tetragonal phase

5. Summary

Pseudo-potentials (Oxygen, Hafnium)

Troullier-Martins scheme

Oxygen (1s2 2s2 2p4)

valance electrons: 2s2 2p4

Hafnium ( [Xe]4f 14 5d26s2 )

valance electrons: 5s2 5p6 5dx 6sy

eigenvalue transferability x=3, y=0

non-linear core correction (Rc =0.7 a.u)

ghost states local component = s wave

local component = p wave

Atomic wave function of 5shell for Hf

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 2 4 6

r (a.u)

U(r)

5d5p5s

0

50

100

150

200

0 1 2 3 4

r (a.u)

4p r2 n

(r)

pseudomodel coretrue core

Radial Densities for Hf

Rc

DFT-LDA calculation

The all-state-preconditioned conjugate gradient scheme (CCG) for structural calculationThe state by state conjugate gradient scheme (DIIS_CCG) for band calculation

Ecut = 70Ry

k-points = 4 x 4 x 4 Monkhorst-Pack grid(irreducible k-points=10 and 6 for cubic and tetragonal phase)

Lattice constant of c-HfO2 Bulk modulus of c-HfO2

4.95.05.15.25.3

0 50 100

Ecut (Ry)

aLat

(Å)

0.0100.0200.0300.0400.0

0 50 100

Ecut (Ry)

B0

(GPa

)

SFHIngX (Plane wave basis set)

Structural Properties

DFT-LDA LAPW EXP * Error

a (Å) 4.95 5.00 5.08 2.6% (1.1%)

B0 (GPa) 280 277 no-data (1.1%)

DFT-LDA Exp ** Error

a (Å) 4.94 5.15 4.1%

c (Å) 5.054 5.289 4.4%

c/a 1.023 1.027 0.4%

dz (Å) 0.202 no-data no-data

dz /c 0.040 no-data no-data

B0 (GPa) 274.6 no-data no-data

a

dz

a

c

Structural parameters are in good agreement with experimental values (within 5%)

Cubic phase

Tetragonal phase

*J.Amer.Ceram.Soc.53,264 (1970)

**J.Amer.Ceram.Soc.55,482 (1972)

Electronic Properties (cubic phase)

Top of valance band O 2p stateBottom of conduction band Hf 5d stateBand gap ~ 4eV

Partial density of states (LDA)

0

2

4

6

8

10

-80 -60 -40 -20 0 20Energy (eV)

DO

S (A

rb. u

nits

)

O(s)O(p)Hf (s)Hf (p)Hf (d)

)()()()()]()([ LDALDALDALDAXC

LDAH rrrrrr kkkk nnnn VVh

)(')'(),',()()]()([ QPQP3QPQPQPH rrrrrrr kkkkk nnnnn rdVh

)()(,',)( LDALDAXC

QPLDALDAQP rrrrr kkkkk nnnnn V

QPLDAkk nn

Kohn-Sham equation: (ground state properties)

Quasiparticle equation (GW calculation):

First-order correction:

GW approximation self-energy==iGWG = one-particle Greens function, W = screened Coulomb interaction

GW correction for band structure calculation

Electronic Properties (cubic phase)

kz

ky

kx

b1

b3

b2 L

W

X

K

Band energy (eV)

Eg

DFT-LDA 3.71 eV

LAPW 3.55 eV

LDA +GW 5.51 eV

EXP * 5.8 eV

* Y2O3 (0.15) HfO2 (0.85)

J.Appl.Phys vol, 91 4500 (2002)

GW correction ~1.8 eV

Eg (direct)

LDA+GW

LDA

Electronic Properties (tetragonal phase)kz

ky

kx

b1

RZ

A

Mb2

b3

Eg

DFT-LDA 4.11 eV

LDA +GW 5.82 eV

Band energy (eV)

Band transitionindirect (A to )

GW correction ~1.7 eV

Eg (indirect)

LDA+GW

LDA

c/a factor effect

a=5.15 Å (fixed ), dz/c = 0.0 (fixed)

Band energy (eV)Cubic (c/a=1.00)

c/a=1.027

aa

c

c/a transition Eg

1.00 M to M 3.58

1.01 M to M 3.51

1.02 M to M 3.43

1.027 M to M 3.37

(1) Transition is same

(2) Band gap is decreased

(tetragonal M = cubic X)

Tetra (dz/c=0.04)

c/a=1.027 (dz/c=0.00 )

aa

c

a=5.15 Å (fixed ) , c/a = 1.027 (fixed)

dz/c factor effect

dz/c transition Eg

0.00 M to M 3.37

0.01 M to 3.48

0.02 M to 3.69

0.03 A to 3.89

0.04 A to 4.11

Band energy (eV)

dz/c reflects the differencebetween cubic and tetragonal

(1) DFT-LDA reliability

(2) GW correction

(3) The comparison between cubic and tetragonal phase

Summary

Change from direct to indirect gap is due to internal oxygen relaxation

Cubic phase : LDA+GW (5.5 eV), LDA (3.7eV), experiment (5.8eV)

Structural properties are in good agreement with LAPW and experiment

Band gap is underestimated compared with experiment

Tetragonal phase : LDA+GW (5.8 eV), LDA (4.1eV)

Thank you for your attention

Eigen value transferability test for Hf pseudo-potential

Error = Pseudo - all electron

-100.0

-50.0

0.0

50.0

100.0

0 1 2 3

5d occupancy

Erro

r (m

eV) 5s

5p5d6s

HfHf+

-100.0

-50.0

0.0

50.0

100.0

0 1 2 3

6s occupancy

Erro

r (m

eV) 5s

5p5d6s

5d occupancy

6s occupancy

HfHf+

Other theoretical calculations

Author Paper Hf pp pp Exc content

Xinyuan Zhao PRB vol65, 233106 (2002) 5s 5p 5d 6s Ultrasoft LDA/GGA Structure, epsilonDemkov Phys.Stat.sol 226, 57(2001) no-data CASTEP LDA SrTiO3, HfO2, ZrO2, structure,bandDabrowski MR vol41, 1093 (2001) no-data norm-conserving LDA/GW GW band structure, Pr psuedopotentialFiorentini PRL vol 89, 266101 (2002) semi-core VASP GGA Band offset, formatin energy, epsilonNieminen group PRB vol65, 174117 (2002) 5d(3) 6s(1) VASP GGA Vacancy, interstial defectJoongoo Kang PRB vol68, 054106 (2003) no-data norm-conserving LDA/GGA Structure, phase transitionRignanese PRB vol69, 184301 (2004) 5s 5p 5d 6s norm-conserving LDA Silicates, structure,epsilonLowther PRB vol60, 14485 (1999) no-data norm-conserving LDA StructureRobertson group PRB vol92, 057601 (2004) no-data CASTEP LDA ZrO2 Interface, Band offsetRobertson group APL vol84, 106 (2003) no-data CASTEP LDA HfSiON, Band offsetRobertson group JAP vol92, 4712 (2002) no-data CASTEP LDA many High-K, Band offsetRobertson group JVST.B18, 1785 (2000) no-data CASTEP LDA many High-K, Band offsetJomard PRB vol59, 4044 (1999) semi-core Ultrasoft LDA ZrO2, structureFinnis PRL vol81, 5149 (1998) no-data no-data no-data ZrO2, structureKralik PRB vol57, 7027 (1998) semi-core norm-conserving LDA/GW ZrO2, GW band structureFrench PRB vol49, 5133 (1994) no-data no-data no-data ZrO2, band structureParlinski PRL vol78, 4063 (1997) no-data CASTEP no-data ZrO2, structure, phonon dispersion

Band gap vs lattice constant (cubic phase)

a (Å) Eg (eV)4.80 4.214.90 4.024.95 3.955.00 3.855.08 3.715.10 3.675.20 3.51 3.00

3.203.403.603.804.004.204.40

4.70 4.80 4.90 5.00 5.10 5.20 5.30aLat (Å)

Eg (e

V)

Comparison between cubic and tetragonal phasekz

ky

kx

b1

RZ

A

Mb2

b3

Band energy (eV)

(tetragonal M = cubic X)

transition Eg

cubic M to M 3.58

tetra A to 4.11

(1) Transition is different

(2) Band gap is increasing

Tetragonal

Cubic (a=5.15Å

Eg (direct)

Band transition change (tetragonal phase)

c/a=1.00 VB CBA 1.9603 6.7306M 2.1542 5.7345G 2.1540 5.7342

c/a=1.01 VB CBA 1.9460 6.5721M 2.1287 5.6377G 2.0219 5.6450

c/a=1.02 VB CBA 1.9279 6.4090M 2.0992 5.5300G 1.8925 5.5496

c/a=1.027 VB CBA 1.9147 6.2951M 2.0787 5.4508G 1.8056 5.4648

dz/c=0.00 VB CBA 1.9147 6.2951M 2.0787 5.4508G 1.8056 5.4648

dz/c=0.01 VB CBA 1.8847 6.3326M 2.0102 5.5154G 1.7444 5.4911

dz/c=0.02 VB CBA 1.7986 6.4441M 1.8211 5.6965G 1.5768 5.5156

dz/c=0.03 VB CBA 1.6688 6.6253M 1.5496 5.9643G 1.4537 5.5592

dz/c=0.04 VB CBA 1.4967 6.8532M 1.2179 6.281G 1.3847 5.611

GW calculation

<Convergence Parameter>

Number of empty states = 800 states for correlation part

k points = 4 x 4 x 4

ecut off = 36Ha / 20Ha for cubic (exchange / correlation)

36 Ha / 24Ha for tetragonal (exchange /correlation)

GWST

Space-time method

Space - time method

Hedin`s GW approximation Space-time method*

k k

kk rrrrn n

nn

iG

LDALDA

;,

;,;,;, rrrrrr GGdP

;,;, 3 rrrrrrrr Prd

;,;, 13 rrrrrr rdW

ieGWd ;,;,;, rrrrrr

LDA LDA*n

unoccLDA exp kkk rr n

nni

LDAn

LDA*n

occLDAn exp;, kkkk rrrr

n

iiG

iGiGiP ;,;,;,0 rrrrrr RRR

convolutions

GGGGGG kG

kGGk

,;,;, iPi

iiW ;,,;, 1 GGGGGGG

kkk

iWiGi ;,;,;, rrrrrr RRR

multiplications

FFT

FFT

*Rieger et al. CPC 117, 211-228 (1999)

real space, energy domainreal space reciprocal space, imaginary time

LAPW method

(1) inside atomic sphere

l max =10 Hf (l =0, 1, 2 : APW+lo, l>2 : LAPW)O (l = 0, 1 : APW+lo, l>1 : LAPW)

Muffin tin radius ( Hf = 2.0 a.u, O =1.7 a.u )

(2) interstitial region

Plane wave cut off = 21.1Ry

Wien 2K

Hf atom energy level

n l occupation Enl(eV)1 0 2 -65232.72 0 2 -11157.12 1 6 -9800.43 0 2 -2541.63 1 6 -2136.43 2 10 -1655.94 0 2 -511.44 1 6 -378.44 2 10 -206.54 3 14 -17.05 0 2 -67.25 1 6 -35.75 2 2 -2.96 0 2 -5.3

Energy level

-2.9eV-5.3eV

-35.7eV

-67.2eV

-17.0eV

5d6s

4f

5p

5s

fhi98pp- program

Anisotropy in tetragonal phase

The head of dielectric matrix

iw dir1 dir2 dir3 anisotropy(%)0.0318 6.29 6.29 6.04 4.190.1663 5.07 5.07 4.94 2.670.4031 3.14 3.14 3.11 1.080.7338 2.07 2.07 2.06 0.441.1464 1.58 1.58 1.57 0.181.6259 1.34 1.34 1.34 0.072.1552 1.21 1.21 1.21 0.032.7150 1.14 1.14 1.14 0.023.2850 1.10 1.10 1.10 0.013.8448 1.08 1.08 1.08 0.004.3741 1.06 1.06 1.06 0.004.8536 1.05 1.05 1.05 0.005.2662 1.04 1.04 1.04 0.005.5969 1.04 1.04 1.04 0.005.8337 1.04 1.04 1.04 0.005.9682 1.03 1.03 1.03 0.007.7587 1.02 1.02 1.02 0.00

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10

0 5 10

iwan

isot

ropy

(%)